Find the indefinite integral of (x^4 - 3x^2 + 5)/x with respect to x.

Step-by-step solution

1. 1

   Do not reach for substitution. A single monomial denominator means the fraction splits cleanly, and splitting first turns one awkward quotient into three standard integrals. Substitution or parts would both be wasted effort here.

2. 2

   Divide each numerator term by $x$. $\frac{x^4-3x^2+5}{x}=\frac{x^4}{x}-\frac{3x^2}{x}+\frac5x=x^3-3x+\frac5x.$

3. 3

   Apply the power rule to the polynomial part. $\displaystyle\int x^3\,dx=\frac{x^4}{4}$ and $\displaystyle\int -3x\,dx=-\frac{3x^2}{2}$.

4. 4

   Handle the $5/x$ term with the log rule. The power rule fails at exponent $-1$, so instead $\displaystyle\int\frac5x\,dx=5\ln|x|$. The absolute value matters because the original integrand is defined for negative $x$ too.

5. 5

   Assemble and verify by differentiating. $\int\frac{x^4-3x^2+5}{x}\,dx=\frac{x^4}{4}-\frac{3x^2}{2}+5\ln|x|+C.$ Differentiating gives $x^3-3x+\frac5x$, which is exactly the simplified integrand, so the answer checks out.